An integral characterizing the Andreotti-Mayer locus
The Andreotti-Mayer locus in the moduli space \({\mathcal A}_n\) of principally-polarised abelian \(n\)-folds is the locus \({\mathcal N}_0\) for which the theta divisor is singular. It is a divisor and it is given by the vanishing of a certain Siegel modular form \(F_n\) of weight \(k=n!(n+3)/2\). The present paper exhibits a current \(g\) on the universal family \(A\) whose direct image \(p_g\) in \({\mathcal A}_n\) is of type \((0,0)\) and satisfies \[ dd^c(p_*g)+\delta_{{\mathcal N}_0}=c_1({\overline\omega}^{\otimes k}), \] where \(\overline\omega\) is \(\det p_*\Omega^1_{A/{\mathcal A}_n}\) equipped with the Petersson metric \(\|\cdot\|_{\text{Pet}}\). For a suitable multiple \(\Delta_n\) of \(F_n\) one has \(p_*g=-\log\|\Delta_n\|_{\text{ Pet}}^2\). From this the authors deduce an integral formula for \(\log\|\Delta_n\|_{\text{Pet}}^2\) in terms of suitably metrised line bundles on \({\mathcal A}_n\) and on the universal theta divisor. An equivalent formula was found by different methods by \textit{K. Yoshikawa} [J. Differ. Geom. 52, No. 1, 73-115 (1999; Zbl 1033.58029)]. The current \(g\) and its properties are derived from Green's currents on the metrised line bundles.
scientific article; zbMATH DE number 1868565
scientific article